Introduction

This paper treats bicycles from the perspective of control. Models of differentcomplexity are presented starting with simple ones and ending with more real-istic models generated from multibody software. Models that capture essentialbehavior such as self-stabilization and demonstrate the dif?culties with rearwheel steering are presented. Experiences of using bicycles in control educationare presented with suggestions for fun and thought-provoking experiments withproven student attraction. The paper also describes design of adapted bicyclesfor children with disabilities and clinical experiences of their use.The bicycle is used everywhere, for transportation, exercise, and recreation.The bicycle’s evolution over time was a product of necessity, ingenuity, materi-als, industrialization, invention, imagination, and yet with comparatively littletheoretical insight. The bicycle is ef?cient, highly maneuverable, and yet it rep-resents a tantalizing enigma. Learning to ride a bicycle is an acquired skill,often acquired with some dif?culty, and yet once mastered the skill becomes subconscious and second nature, literally just "as easy as riding a bike."Bicycles have interesting dynamic behavior. They are statically unstable, likeinverted pendulums, but can under certain conditions be stable in forward mo-tion. They also exhibit non-minimum phase steering behavior. There are alsononlinearities due to geometry and tire-road interactions.Bicycles have intrigued scientists ever since they appeared in the middle ofthe nineteenth century and there is a considerable literature on bicycles. Thebook by Sharp is a classic from 1896 that has recently been reprinted [1], thebooks [2], [3] give a broad engineering perspective. There are early papers fromthe 19th century [4] - [8]. Famous scientists like Rankine [4], Klein and Som-merfeld have analyzed bicycles [9]. It is notable that Klein and Sommerfeld wereparticularly interested in the effect of gyroscopic action of the front wheel. Pa-pers on bicycles appear regularly in literature [10] - [22]. The ?rst publicationsof differential equations describing the motion of an idealized bicycle appearedtowards the end of the 19th century. Notable are Whipple [7], Carvallo [8], [23],who derived equations of motion, linearized around the vertical equilibrium con-?guration, from Lagranges equations. During the early 20th century severalauthors studied the problems of bicycle self stability, balancing, and steering thebicycle, with variable success. The paper [24] presented one of the ?rst computersimulations of a nonlinear bicycle model. Neˇ?mark and Fufaev [25] derived acomprehensive set of linear models by approximating potential and kinetic en-ergy by quadratic terms and applying Lagrange’s equations to these expressions.They used different wheel models, ideal disks as well as pneumatic tires. Theirmodel is elaborated in the book [26]. Modeling of bicycles became a popular topicfor dissertations in the later half of the last century, [27] - [33]. Simple models ofbicycle dynamics are given in the text books [34], [35] and [36]. Nonlinear modelsare presented in [21], [37], [38] and [39].#p#分頁標題#e#

Geometry and Coordinate Systems

Simple Second Order Models
Rear-wheel Steering
Bicycles in Education
ManeuveringBicycles in EducationAdapted Bicycles for Teaching Children with Disabilities
More Complicated Models
Conclusions

1. References

[1] S. Archibald.Bicycles & Tricycles : A Classic Treatise on Their Design andConstruction. Dover, New York, 2003. Dover publication of classic from 1896.

[2] F. R. Whitt and D. G. Wilson.Bicycling Science. MIT Press, Cambridge, MA,1982. Second edition.

[3] D. G. Wilson.Bicycling Science. MIT Press, Cambridge, MA, 2004. Thirdedition, with contributions by Jim Papadopoulos.

[4] M. W. J. Rankine. “On the dynamical principles of the motion of velocipedes.”The Engineer, 28, pp. 79, 129, 153, 157, 1869.

[5] C. Bourlett.Traite’ des bicycles et Bicyclettes. Gauthier-Villars, Paris, 1898.

[6] S. Achibald.Bicycles and tricycles, an elementary treatise on their designand construction. Longmans Greens and Co, London, 1896.

[7] F. J. W. Whipple. “The stability of the motion of a bicycle.”Quarterly Journalof Pure and Applied Math., 30, pp. 312–348, 1899.

[8] Carvallo. “Théorie du mouvement du monocycle et de la bicyclette.”Journalde L’Ecole Polytechnique, 5, 1900. 119-188.

[9] F. Klein and A. Sommerfeld.Über die Theorie des Kreisels, vol. Heft IV.Teubner, Leipzig, 1910. pp. 863–884.

[10] A. M. Letov. “Stability of an automatically controlled bicycle moving on anhorizontal plane.”Applied Mathematics and Mechanics, 23:4, pp. 650–655,1959.37

[11] H. Fu. “Fundamental characteristics of single-track vehicles in steadyturning.”Bull. Japanese Soc. of Mech. Eng., 9, pp. 284–293, 1965. checkjournal title JSME Bulletin. Check year 1965 or 1966.

[12] D. E. H. Jones. “The stability of the bicycle.”Physics Today, April, pp. 34–40, 1970. The notion of URB.

[13] A. I. Krauter. “Steady-state cornering of two-wheeled vehicles.”J. Appl.Mechanics, September, pp. 819–821, 1973.

[14] T. R. Kane. “Fundamental kinematical relationships for single-track vehi-cles.”Int. J. Mechanical Sciences, 17, pp. 499–504, 1975.

[15] J. Liesegang and A. R. Lee. “Dynamics of a bicycle: non-gyroscopic effects.”American J. of Physics, 46, pp. 130–132, 1978.

[16] D. H. Weir and J. W. Zellner. “Lateral directional motorcycle dynamics andcontrol.” InAmerican Society of Automotive Engineers. American Society ofAutomotive Engineers, 1978. SAE Paper 780304.

[17] D. Kirshner. “Some non-explanations of bicycle stability.”Am. J. Physics,48:1, pp. 36–38, 1980.

[18] M. Kondo. “Experimental study on the stability and control of single-trackvehicles.”Proc. Japan Society of Mechanical Engineering, 58, pp. 827–833,1955. Check Journal title.

[19] J. Lowell and H. D. McKell. “The stability of bicycles.”Am. J. Phys., 50:12,pp. 1106–1112, 1982.[20] Y. Le Henaff. “Dynamical stability of the bicycle.”European J. of Physics,8, pp. 207–210, 1987.38[21] G. Franke, W. Suhr, and F. Rieß. “An advanced model of bicycle dynamics.”European Journal of Physics, 11:2, pp. 116–121, 1990.[22] A. L. Schwab, J. P. Meijaard, and J. M. Papadopoulos. “Benchmark resultson the linearized equations of motion of an uncontrolled bicycle.” InProc.Second Asean Conference on Multibody Dynamics, August 1-4,2004, Seoul,Korea, 2004.[23] Carvallo. “Théorie du movement du monocycle. part 2: Theorie de labicyclette.”Journal d l’Ecole Polytechnique, 6, 1901. 1-118.[24] R. D. Roland.Computer simulation of bicycle dynamics. ASME - Mechanicsand Sport, 1973.[25] WIU. I. Neˇ?mark and N. A. Fufaev. “The stability of noncontrolled andcontrolled bicycles and motorcycle.”Journal of Mechanics of Solids, 2:2,pp. 7–14, 1967.[26] WIU. I. Neˇ?mark and N. A. Fufaev.Dynamics of Nonholonomic Systems.American Mathematical Society, 1972. ISBN 0-8218-1583-0.[27] D. V. Singh.Advanced Concepts of the Stability of Two-wheeledVehicles-ApplicationofMathematicalAnalysistoActualVehicles. PhD thesis,University of Wisconsin, Madison, WI, 1964.[28] R. S. Hand.Comparisons and stability analysis of linearized equations ofmotion for a basic bicycle model. Masters thesis, Cornell University, Ithaca,NY, 1988.[29] D. H. Weir.Motorcycle Handling Dynamics and Rider Control and theEffect of Design Con?guration on Response and Performance. PhD thesis,39University of California, Los Angeles, Los Angeles, CA, 1972.[30] D. J. Eaton.Man-MachineDynamicsintheStabilizationofSingle-TrackVehicles. PhD thesis, University of Michigan, Ann Arbor, MI, 1973.[31] L. H. Laiterman.Theory and applications of rear-wheelsteeringinthedesign of man-poweredlandvehicles. BS thesis, MIT, Cambridge, MA, 1977.[32] C. Koenen.The dynamic behaviour of a motorcycle when running straightahead and when cornering. PhD thesis, Technische Hogeschool Delft, Delft,The Netherlands, 1983.[33] B. C. Mears.Open loop aspects of two wheeled vehicle stability character-istics. PhD thesis, University of Illinois, Urbana-Champaign, IL, 1988.[34] S. Timoshenko and D. H. Young.Advanced Dynamics. McGraw Hill, NewYork, 1948. page 239.[35] J. P. Den Hartog.Mechanics. Dover, New York, 1961. page 328, ?rst issuedby Mc Graw Hill 1948.[36] A. E. Bryson.Applied Linear Optimal Control: Examples and Algorithms.Cambridge University Press, 2002.[37] N. H. Getz. “Control of balance for a nonlinear nonholonomic non-minimumphase model of a bicycle.” InProc. ACC, pp. 148–151, Baltimore, MD, 1994.[38] N. H. Getz and J. E. Marsden. “Control for an autonomous bicycle.” InProc.IEEE Conf. Robotics and Control, Nagoya, Japan, 1995.[39] W. S. Koon and J. E. Marsden. “The hamiltonian and lagrangian ap-proaches to the dynamics of nonholonomic systems.”Reports in Mathemat-40ical Physics, 40, pp. 21–62, 2000.[40] R. E. Roberson and R. Schwertassek.Dynamics of Multibody Systems.Springer, Berlin, 1988.[41] R. Von Schwerin.Multibody System Simulation: Numerical Methods, Algo-rithmsandSoftware. Springer, Berlin, 2000.[42] M. Coutinho.Dynamic Simulation of Multibody Systems. Springer, Berlin,2001.[43] A. Ahmed.Dynamics of Multibody Systems. Cambridge University Press,Cambridge, 2003.[44] A. Lennartsson.Ef?cient multibody dynamics. PhD thesis, Royal Instituteof Technology, Stockholm, Sweden, 1999.[45] A. L. Schwab.Dynamics of ?exible multibody systems. PhD thesis, DelftUniversity of Technology, Delft, Netherlands, 2002.[46] R. S. Sharp. “The stability and control of motorcycles.”J. Mech. Engrg. Sci.,13:5, pp. 316–329, 1971.[47] R. S. Sharp. “The dynamics of single track vehicles.”Vehicle SystemDynamics, 5, pp. 67–77, 1975.[48] H. B. Pacejka.Tyre and Vehicle Dynamics. Butterworth, Oxford, 2002.[49] P. A. J. Ruijs and H. B. Pacejka. “Research in the lateral dynamics ofmotorcycles.” InProc. 9th IAVSD Symposium on the Dynamics of Vehicleson Roads and Tracks., pp. 467–478. International Federation of AutomaticControl, 1996.41[50] V. N. Fomin, A. L. Fradkov, and V. A. Yakubovich.Adaptive control ofdynamics plants. Nauka, Moscow, 1981. in Russian pp. 14–22 and pp. 257–268.[51] S. Lee and W. Ham. “Self stabilizing strategy in tracking control ofan unmanned electric bicycle with mass balance.” InProc. IEEE Conf.Intelligent Robots and Systems, pp. 2200–2205, Lausanne, Switzerland,2002.[52] Y. Tanaka and T. Murakami. “Self sustaining bicycle robot with steeringcontroller.” InProc. IEEE Int. Workshop on Advanced Motion Control AMC2004, pp. 193–197, Kawasaki, Japan, 2004.[53] H. Goldstein.Classical Mechanics. Addison-Wesley, Cambridge, MA, 1953.p. 145.[54] S. Skogestad and I. Postlethwaite.Multivariable feedback control: analysisand design. Wiley, Chichester, UK, 1996.[55] K. J. Åström. “Limitations on control system performance.”EuropeanJournal on Control, 6:1, pp. 2–20, 2000.[56] J. M. Papadopoulos. “Bicycle steering dynamics and self-stability: A sum-mary report on work in progress.” Technical Report, Cornell Bicycle Project.Cornell University, Itaca, NY, 1987. Unpublished notes.[57] R. E. Klein. “Novel systems and dynamics teaching techniques usingbicycles.” InPreprints Joint American Control Conference(JACC), vol. 2,pp. 1157–1160, Atlanta, GA,#p#分頁標題#e# 1988.42[58] R. E. Klein. “Using bicycles to teach system dynamics.”IEEE ControlSystems Magazine, 6:4, pp. 4–9, 1989.[59] R. E. Klein. “The University of Illinois bicycle project.” InIFAC Workshopon Advances in Control Education, Boston MA, June 1991.InternationalFederation of Automatic Control, 1992.[60] K. J. Åström and J. Lunze. “Warum können wir Fahrrad fahren.”Automa-tisierungstechnik, 49, pp. 427–435, 2001.[61] R. E. Klein, L. Lieberman, P. DiRocco, and E. McHugh. “Adapted bikesdeliver new independence.”Exceptional Parent, 32:10, pp. 64–66, 2002.[62] R. E. Klein, E. McHugh, S. L. Harrington, T. Davis, and L. Lieberman.“Adapted bicycles for teaching riding skills.”Teaching Exceptional Children,2004. submitted for publication.[63] G. Martin and J. Pear.Behavior modi?cation; What it is and how to use it.Prentice Hall, Upper Saddle River, NJ, 2003. page 123.[64] W. S. Lee, B. D. O. Anderson, I. M. Y. Mareels, and R. Kosut. “On some keyissues in the windsurfer approach to adaptive robust control.”Automatica,31:11, pp. 1619–1636, 1995.[65] H. Elmqvist, S. E. Mattsson, and M. Otter. “Modelica a language forphysical system modeling, visualization and interaction.” InThe 1999 IEEESymposium on Computer-AidedControlSystemDesign,CACSD’99,Hawaii,August 22-27,1999, 1999. See also http://www.modelica.org.[66] M. Otter, H. Elmqvist, and S. Mattsson. “The new modelica multibodylibrary.” InProceedings of the 3rd International Modelica Conference,43Linköping, November 3-4,2003, pp. 311–330. Modelica, 2003. Downloadfrom: http://www.modelica.org/Conference2003/papers/h37_Otter_multibody.pdf.[67] M. Lesser.The Analysis of Complex Nonlinear Mechanical Systems, aComputer Algebra Assisted Approach, vol. 17 ofWorld Scienti?c Series onNonlinear Science-SeriesA. World Scienti?c, Singapore, 1995. ISBN 981-02-2209-2.[68] A. Sommerfeld.Mechanics. Academic Press, New York, 1952.[69] K. I. T. Richardson.The Gyroscope Applied. Hutchinson, London, 1954.[70] R. M. Murray, K. J. Åström, S. Boyd, R. W. Brockett, and G. Stein. “Futuredirections in control in an information-rich world.”IEEE Control SystemsMagazine, 23:2, pp. 20–33, 2003.[71] R. M. Murray.Control in an information rich world-ReportofthePanelonFuture Directions in Control, Dynamics, and Systems. SIAM, 2003. ISBN0-89871-528-8.[72] R. Schwarz. “Accident avoidance characteristics of unconventional motor-cycle con?gurations.”SAE Paper 790258, Society of Automotive Engineers,Warrendale, PA, 1979.[73] G. Stein. “Respect the unstable.”IEEE Control Systems Magazine, 23:4,pp. 12–25, 2003.44Side-bar 1 - The NHSA Rear-Steered Motorcycle